Let us now return to the given inequality.
If t=0, the claim is trivially true for all x,y in ℝn.
If t>0, then raising both sides in inequality 1 to
the power of t, using t=|t|, and setting a=x, b=y yields the result.
On the other hand, if t<0, then raising both sides in inequality
1 to the power to -t, using -t=|t|, and setting
a=y, b=x yields the result.
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